arthimetic mean of a and b
Geometry
Level
2
If (a,b) is circumcentre of atriangle the combined equation of whose sides is (x+y-4)(xy-2x-y+2)=0,find arthimetic mean of a and b.
The answer is 2.
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1 solution
induvidual equations of sides are(1)x+y-4=0;(2)x-1=0;(3)y-2=0; these equations form right angled triangle; note:circumcentre of right angled triangle is mid point of hypotenuse. now in this triangle hypotenuse is x+y-4=0 this will be clear if you draw graph. point of intersection of x+y-4=0 and x-1=0 is (1,3) and for x+y-4 and y-2=0 is (2,2)circumcenter=mid point of hypotenuse =(1.5,2.5)=(a,b) arithmetic mean=(a+b)/2 =(1.5+2.5)/2 =2